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Causality and dielectric functions for linear media with spatial dispersion

Josep Llosa, Francesc Salvat

Abstract

We extend Kramers-Kronig relations beyond the optical approximation to dielectric functions that depend not only on frequency but on the wave number as well. This implies extending the notion of causality commonly used in the theory of Kramers-Kronig relations to include the fact that signals cannot propagate faster than light in vacuo. The extension is applied to some microscopic models for the dielectric function and is compared with previous generalizations. The results derived here also apply to general theories of isotropic linear response in which the response function depends on both wave number and frequency.

Causality and dielectric functions for linear media with spatial dispersion

Abstract

We extend Kramers-Kronig relations beyond the optical approximation to dielectric functions that depend not only on frequency but on the wave number as well. This implies extending the notion of causality commonly used in the theory of Kramers-Kronig relations to include the fact that signals cannot propagate faster than light in vacuo. The extension is applied to some microscopic models for the dielectric function and is compared with previous generalizations. The results derived here also apply to general theories of isotropic linear response in which the response function depends on both wave number and frequency.

Paper Structure

This paper contains 21 sections, 1 theorem, 116 equations.

Table of Contents

  1. Introduction
  2. Consequences of finite speed causality on the susceptibility function
  3. Symmetries
  4. The susceptibility function at high frequencies
  5. The existence of the Laplace transform $g(s_+,s_-)$
  6. The optical approximation
  7. A generalization of Kramers-Krönig relations
  8. Hilbert transform:
  9. The standard Kramers-Krönig relations
  10. The significance and usefulness of causality conditions
  11. Application to some dielectric functions
  12. Degenerate electron gas. Semiclassical model
  13. Non-relativistic quantum degenerate electron gas (random phase approximation)
  14. Valence electrons in semiconductors and insulators
  15. Relativistic quantum degenerate electron gas
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\, g(s_+,s_-)\,$ be the double Laplace integral (e9z), assume that $\,G(u^+,u^-)\,$ has continuous partial derivatives $\,G^{(l,j)}(u^+,u^-) = \partial_+^l \partial_-^j G(u^+,u^-)\,$ up to the $N$-th order (i. e. $l+j\leq N$) and that there exist $\,a_\pm \in \mathbb{R} \,$ such that $\;\int_{\

Theorems & Definitions (1)

  • Theorem 1